Computational Matrix Representation Modules for Linear Operators With Explicit Constructions for a Class of Lie Operators
Document Type
Article
Publication Date
10-5-1998
Identifier/URL
40912294 (Pure)
Abstract
A linear mapping from a finite-dimensional linear space toanother has a matrix representation. Certain multilinear functions arealso matrix-representable. Using these representations, symboliccomputations can be done numerically and hence more efficiently. Thispaper presents an organized procedure for constructing matrixrepresentations for a class of linear operators on finite-dimensionalspaces. First we present serial number functions for locating basismonomials in the linear space of homogeneous polynomials of fixeddegree, ordered under structured lexicographies. Next, basic lemmasdescribing the modular structure of matrix representations foroperators constructed canonically from elementary operators arepresented. Using these results, explicit matrix representations arethen given for the Lie derivative and Lie-Poisson bracket operatorsdefined on spaces of homogeneous polynomials. In particular, they arecomprised of blocks obtained as Kronecker sums of modular components,each corresponding to specific Jordan blocks. At an implementationlevel, recursive programming is applied to construct these modularcomponents explicitly. The results are also applied to computing powerseries approximations for the center manifold of a dynamicalsystem. In this setting, the linear operator of interest isparametrized by two matrices, a generalization of the Lie-Poisson bracket.
Repository Citation
Fu, J. H.,
& Miller, D. F.
(1998). Computational Matrix Representation Modules for Linear Operators With Explicit Constructions for a Class of Lie Operators. Journal of Computational and Applied Mathematics, 98 (1), 1-26.
https://corescholar.libraries.wright.edu/math/508
DOI
10.1016/S0377-0427(98)00088-0
